A NEW CHARACTERISTIC SUBGROUP "

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M'iCHiGAN STATE UNIVERSITY
STEPHEN F .. MARKSTEIN
1973

 

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A NEW CHARACTERISTIC SUBGROUP
OF INFINITE GROUPS

presented by

Stephen F. Markstein
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ABSTRACT

A NEW CHARACTERISTIC SUBGROUP
OF INFINITE GROUPS

BY
Stephen F. Markstein

In this dissertation we investigate a new characteristic
subgroup, S(G), of an arbitrary infinite group G. S(G) is
defined by

S(G)={xelehenever H is a non-finitely generated

subgroup of G so is <H,x>}.

S(G) is always a subgroup of the locally Notherian radical,
N(G). Let SN be the class of all groups for which S(G)=N(G).
A subgroup HcG is nearly finitely generated in G if H is not
finitely generated but every subgroup of G properly contain-

ing H is finitely generated, and I(G)=n(HcG|H is nearly

finitely generated in G}.

In Chapter II we develop the basic prOperties of S(G). We
prove, among others:

Lemma: S(G)cI(G).

 

Lemma: All the subgroups of G are S groups if and only if

 

N
N(K)cI(K) for all subgroups K of G.

Corollary: If X is a subgroup-closed class of groups, then

 

XCSN if and only if N(G)CI(G) for all GeX.

Stephen F. Markstein

Lemma: Normal subgroups of 5 groups are SN groups.

IV
In Chapter III we develop further prOperties of 5(6) and the
nearly finitely generated subgroups of G, and we prove our
main results.

Theorem: The class of extensions of Abelian groups by
polycyclic groups is contained in SN'
Definition: A class X satisfies (*) if whenever G is an ex-

 

tension of the X group K by the nilpotent group G/K, then
N(K)cH for all the nearly finitely generated subgroups H of
G.

Theorem: If X is a subgroup-closed class of groups, then
the class of extensions of X groups by nilpotent groups is

contained in S if and only if X satisfies (*).

N

Theorem: Finite extensions of SN groups aretavgroups.

In Chapter IV we construct groups with remarkable pr0perties.
We show how to construct groups G for which S(G)#N(G), and
one of these examples is of a group that is an extension of
a nilpotent group of class 2 by an Abelian group. We also
construct groups having non-normal nearly finitely generated
subgroups, and we show that a nearly finitely generated
subgroup H of G need not intersect a normal subgroup K of
finite index in G in a nearly finitely generated subgroup of

K.

A NEW CHARACTERISTIC SUBGROUP

OF INFINITE GROUPS

BY
Stephen F. Markstein

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1973

......
r

ACKNOWLEDGMENTS

I thank Mr. Kenneth K. Hickin for many fruitful discussions;
the proof of Lemma 3.1 is due to him and it was he who first
noticed PrOposition 2.4. I am grateful to Prof. James E.
Roseblade, who pointed out that enough information was at
hand to prove Theorem 3.10. I am thankful to Prof. Richard
IE. Phillips for reading preliminary versions of this

dissertation and for the useful suggestions he offered.

I am extremely grateful to Prof. Lee. M. Sonneborn, my
advisor, for his moral and mathematical support in the
preparation of this dissertation, as well as throughout my
graduate training, and most particularly for his help in

constructing the examples of Chapter IV.

ii

INTRODUCTION
CHAPTER I.
CHAPTER II.
CHAPTER III.
CHAPTER IV.

BIBLIOGRAPHY

TABLE OF CONTENTS

BACKGROUND 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O 0
PRELIMINARY RESULTS 0 O O O O O O O O O O O O O O O O O O O
THEOREMS O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O I

EXAMPLESOOOOOOOOOOOOOOOOOO0.0.0.0000...

iii

17
31
48

INTRODUCTION

In this paper we introduce a new canonical subgroup, the
S-subgroup, of an arbitrary group G. The inspiration for
our study stems from R. Baer's paper [2], "Group theoretical
pr0perties and functions," and J.B. Riles' paperllfl], "The

near Frattini subgroups of infinite groups."

The primary objective of Baer's paper is the investigation
of possible relationships between group theoretical prOper-
ties and functions. Suppose that X is any isomorphism-
closed class of groups that contains the trivial group, with
the convention that G is an X-group if and only if GeX.

Baer defines the X-hypercenter of an arbitrary group G by

hX(G)=lXeGIHcG, HeX implies <H,x>cKeX for some KcG I.
It is easy to show that hX(G)<>G for all such classes X and

groups G.

Riles starts with a relative pr0perty and defines the set of
non-near generators, U(G), of a group G. U(G) is given by

U(G)=[ xeGI IG:<S> |=°° implies IG:<S,x> I=eo } ,

and it is easy to see that U(G)0 G.

If we alter Baer's definition to
1

2
HX(G)=( xeGlHCG, HeX implies <H,x>exl ,

then without additional assumptions we cannot conclude that
HX(G) is a subgroup of G. However if, for example, we re-
strict ourselves to classes X which are subgroup—closed,

then this definition is equivalent to Baer's.

But it is by no means necessary that X be subgroup-closed

for HX(G) to be a subgroup. Indeed, the S subgroup of G,

the subgroup that we shall study in this dissertation, is ob-

tained by using for X the class of all non-finitely genera-

ted groups, which is obviously not subgroup-closed. Thus,
S(G)=l:xeG|whenever H is a non-finitely generated

subgroup of G so is <H,x>} .

Our primary efforts are aimed at studying the relationship

between S(G) and the locally Notherian radical, ”(G), of G.

Chapter I deals with notation, definitions and the statement
of results from the basic group theory that will be used,
for the most part, without proof or comment. In Chapter II
we prove the basic facts about S(G) and relate S(G) to some
of the well known and intensively studied subgroups of a

group G.

Chapter III contains our main results. We prove that all
groups G which are polycyclic extensions of Abelian groups

have the property that S(G)=N(G), that all finite extensions

3
of groups G for which S(G)=N(G) are again such groups, and
we characterize the subgroup-closed classes X with the
prOperty that the extensions, G, of X groups by nilpotent
groups are such that S(G)=N(G). We also give examples of

such classes X.

In Chapter IV we present examples and constructions dealing
with S(G) and other related subgroups of a group G. In
particular, two of our examples are groups for which

S(G)#N(G).

CHAPTER I

BACKGROUND

In this chapter we list our notation and define the usual
group theory terms as we shall use them. InterSpersed with
the definitions are standard theorems and facts about groups,
whose proofs may be found in [6] or [111. We delay the dis-
cussion of Split extensions, wreath products and relatively

free groups until Chapter IV, where they are needed.

For the remainder of this chapter let G be a group, H, K and
LO, peR, subgroups of G, x and y elements of G, S and T non-

empty subsets of G.

e, 1 or 0 the identity of G
E the identity subgroup of G
<S> the subgroup generated by S
xy y-lxy, the conjugate of x by y
Sx [ sxlseS} , the conjugate of S by x
H h .
S <S |heH>, the normal closure of S in <S,H>
[y,x] y"1x-1yx, the commutator of y and x
[S,T] <[s,t]lseS, teT>
G' [G,G], the commutator subgroup of G

4

Z(G) {zeGlzx=xz for all xeG }, the center of G
91' [stlseS, teT}
Hx [hxlheH }, the right coset of H in G containing x
G/H, g the set of right cosets of H in G
H
|G:H| the cardinality of G/H, the index of H in G
IGI the cardinality of G
2, 7 the group of integers
Zp' 7? the group of integers modulo p
H the empty set
KCG K is a subgroup of G
K<G K is a prOper subgroup of G
K:H K is isomorphic to H

S normalizes K if Kx=K for each xes; K is a normal subgroup

 

 

of G, written K<IG, if G normalizes K; K is characteristic

 

in G, written K<>G, if K=Ka for every automorphism a of G.
Aut(G) is the group of automorphisms of G, and Lat(H,G) is

the lattice of subgroups of G containing H.

If K<IG, then G/K is a group with multiplication given by
(Kx)(Ky)=K(xy). G/K is a factor group, and every right

coset of K in G is also a left coset of K in G. HnK4 H and

H :' HK.

HnK K
If |G:H|<¢ , then H and G are either both finitely generated

or both non-finitely generated.

If C: [LplpeRxH } and if the union of every chain in C is

a member of C, then Zorn's Lemma says that C has maximal

 

members.

A class X of groups is an isomorphism-closed collection of
groups containing E. If X and Y are classes of groups, then

X0! is the class of all extensions of X groups by Y groups;

 

that is, GeXoY if there exists H<:G such that HeX and G/HeY.
.EIKL is the class of all subgroups of members of X. ELXL is
the class of all groups whose finitely generated subgroups
are X groups. A is the class of Abelian groups; flax is the
class of all groups satisfying the maximal condition on sub-
groups, while EEEZB is the class of all groups satisfying

the maximal condition on normal subgroups.

The derived series of G is defined by (i) G(°)=G, and

 

(ii) G(n+l)=[G(n),G(n)], for n20. G is solvable if G‘n)=E

for some integer n. G is polycyclic if there is a series

 

E4 K14 ...4Kn4G with Ki+l/Ki cyclic for each i. G is poly-
cyclic if and only if GeMax and G is solvable. g is the class
of all polycyclic groups. s(P)=P.

The upper central series of G is defined by (i) Z =E, and

0
(ii) Zn+1 is that subgroup of G such that zn+l/zn = 2(G/Zn),

 

n20. ZnOG for each n. The lower central series of G is

 

defined by (i)F1(G)=G, and (ii) [n+1(G)=[G,fn(G)], n21.

7
G is nilpotent if Zn=G for some integer n or equivalently,
rm(G)=E for some integer m. If G is nilpotent, then the
smallest value of n such that Zn=G or equivalently, the
smallest value of m—l such that fm(G)=E, is the gla§§_of G.
Eilp is the class of all nilpotent groups, and Nilp as well

as L(Nilp) are subgroup-closed.

G is a free group if there exists a subset S of G such that

 

every non-identity element geG can be written exactly one

n1 n

way in the form gle . . .xk k, X1! 0 e e ,Xkes, Xizxi+ll

i=1,...,k—l, ni a non-zero integer for each i. In this

 

event S is a set of free generators of G, and §_i§ free 92_

 

S. A word is a member of the free group on the set

S={...,x_l,xo,x1,...}. g 18 the class of all free groups

and s(F)=F. Every group is a homomorphic image of a free

group.

The class L(Max) is the class of locally Notherian groups,
groups whose finitely generated subgroups are Max-groups.
Ifliand K are normal locally N5therian subgroups of G, then
HK is a normal locally N5therian subgroup of G. The locally
N6therian radical 2£.§1 written EIELI is the maximum normal
locally Notherian subgroup of G (see Baer [1] for existence.)
N(G) consists of all elements xeG whose normal closure,

<x>G, is locally Natherian. For every subgroup K of G,

N(G)anN(K).

8

The FC—center of G, Fl(G), is the (characteristic) subgroup

of G consisting of all elements of G with only finitely many

distinct conjugates in G. The ngseries of G is the series

defined by (i) F =E, and (ii) Fn is that subgroup of G

0 +1

 

. . _ 2 . _ . .
satisfying Fn+l/Fn - Fl(G/Fn), n O. G 15 ES nilpotent if

Fn=G for some integer n, and G is an Eg-group if F1(G)=G.

CHAPTER II

PRELIMINARY RESULTS

In the introduction we defined the object of our study, the

S-subgroup of a group G. We recall it here for convenience.

Definition 2.1: S(G): {Xelehenever H is a non-finitely gen-

erated subgroup of G, so is <H,x>.)

Lemma 2.2: S(G) is a characteristic subgroup of G.

Proof: Suppose x,yeS(G), HcG with H not finitely generated,

but <H,xy‘l> is finitely generated. Then <H,x,y> =

<<H,xy‘1>,y> is finitely generated. However, <H,x> is not
finitely generated (since xeS(G)), so that <H,x,y>=<<H,x>,y>
is not finitely generated (since yeS(G)). This contradiction
gives xy'leS(G) so that S(G) is a subgroup of G. That S(G)
is characteristic follows from the fact that automorphisms

take generating sets to generating sets.D

There is another characterization of S(G), which, although
not exploited in this paper, is interesting and included

for that reason.

10

Definition 2.3: Ll(G)= [xeGlH is a finitely generated sub-
group of G implies Lat(H,<H,x>)

eMax },
L2(Gh-( xeGlH is a finitely generated sub-
group of G implies Lat(H,<H,x>)

is finite }.

‘ PrOposition 2.4: S(G)=Ll(G).

 

Proof: Suppose xeS(G), H is a finitely generated subgroup

of G but H<H1<H2<...< <H,x> for some subgroups H1,H2,... of

G. Let K=:Hi. Then K is the union of a prOperly increasing
sequence of groups and cannot be finitely generated. But
<H,x>=<K,x>, so that H is not finitely generated (since
xeS(G)), which is a contradiction. Hence xeL1(G), so
S(G)cL1(G). Conversely, suppose that xeLl(G) and K is a
subgroup of G with <K,x> finitely generated. Then

<K,x>=<k

'00.,kn'x> for some k ,oee'knEK and

1 l

Lat(<kl,...,kn>,<K,x>)eMax since xeLl(G). In particu-

lar, then, all the members of this lattice must be finitely

generated. Since KeLat(<kl,...,kn>,<K,x>), K is finitely

generated so that xeS(G). Thus L1(G)cS(G) and equality
follows.D

11
The proof of Proposition 2.4 is the only argument that we
know of proving that L1(G) is a subgroup of G. Clearly

L2(G) is a subset of L1(G). Whether L2(G) is a subgroup, we

have been unable to decide.
Intimately connected with the study of the S-subgroup of G
are what we term the nearly finitely generated subgroups of

G.

Definition 2.5: If H is a subgroup of G, then H is nearly

 

finitely generated iE,§ if H is not finitely generated, but

 

every subgroup of G prOperly containing H is finitely

generated.

Thus, H is nearly finitely generated in G if and only if
H is maximal in G with reSpect to being non-finitely

generated.

Lemma 2.6: If H is a non-finitely generated subgroup of G,
then H is contained in a nearly finitely generated subgroup

H of G.

N: Let H= chG|HcK, K is not finitely generated) .

Then Hxfl since HeH. If C is a chain in H and C=uC, then C
is not finitely generated since no proper union of groups is
finitely generated. Hence CeH and by Zorn's Lemma H has

maximal members. It is clear that any maximal member H of H

12

is nearly finitely generated in G.D

The intersection of all the nearly finitely generated sub-

groups of G yields another characteristic subgroup of G.

Definition 2.7: I(G)=n [HIH is nearly finitely generated

in G ). (I(G)=G if GeMax.)
Lemma 2.8: S(G) is a subgroup of I(G).

Pgoof: Suppose xeS(G) and H is nearly finitely generated in
G. Then H is not finitely generated and neither is <H,x>
since xeS(G). Hence er since H is maximal non-finitely
generated in G, so S(G)cH, whence S(G)cI(G) since H is an

arbitrary nearly finitely generated subgroup of G.U

Lemma 2.9: S(G) is a subgroup of N(G).

Proof: Suppose xeS(G). We show that <x>G is locally
Notherian. Since S(G)q G and xeS(G), <x>GCS(G). If

G

Kc<x1,...,xn>c<x> and K is not finitely generated, then

adjoining each xi, in turn, to K gives a non-finitely gener-
ated subgroup at each step since xieS(G) for all i, and thus

<xl,...,xn>=<K,xl,...,xn> is not finitely generated, a con-

tradiction. Hence K is finitely generated and <x>GcN(G)

so that S(G)CN(G).U

13
The main positive results in our paper concern classes X
for which GeX implies S(G)=N(G). This suggests the follow-

ing definition.

Definition 2.10: SN is the class of all groups for which

S(G)=N(G).

The following lemma is of value in proving that all sub-

groups of a group G are SN groups.

Lemma 2.11: All subgroups of a group G are SN groups if and

only if for each subgroup K of G, N(K)cI(K).

Proof:+: If all subgroups of G are SN groups, then Lemma

2.8 yields N(K)cI(K) for all subgroups K of G.

+: Suppose that N(K)cI(K) for all subgroups K of G, but
that for some subgroup K and some X€N(K), x¢S(K). Then
there exists a non-finitely generated subgroup H of K such
that <H,x> is finitely generated. By Lemma2.6 there exists
a nearly finitely generated subgroup H of <H,x> that con-
tains H. Then <H,x>=<H,x> and XeN(<H,x>) since X€N(K). By
hypothesis xaH since H is nearly finitely generated in
<H,x>. But this says that <H,x> is not finitely generated,

a contradiction.D

As an immediate corollary of this lemma, we find the groups

that are their own S-subgroups.

l4

Corollary 2.12: S(G)=G if and only if N(G)=G.

Proof: If S(G)=G, then N(G)=G since S(G)cN(G) by Lemma 2.9.
Conversely, if N(G)=G, then I(K)=N(K)=K for all subgroups K

of G so that N(G)=S(G) by Lemma 2.11.0

Another immediate corollary gives a criterion for proving

that a subgroup-closed class is contained in SN'

Corollary 2.13: If X is a subgroup-closed class, then X is

contained in SN if and only if N(G)cI(G) for all GeX.

Of course, in applying Corollary 2.13, we need only worry
about the finitely generated X groups since any non-finitely
generated group trivially satisfies N(G)cI(G)=G. We may
apply this corollary to the class L(X) in the event that X

is contained in SN’

 

Corollary 2.14: If X is a subgroup-closed class of SN groups,

then L(X) is a class of SN groups.

Pgoof: L(X) is subgroup-closed since X is, so, by Corollary
2.13, it suffices to show that N(G)cH whenever H is a nearly
finitely generated subgroup of the L(X) group G. Since we

need concern ourselves only with the finitely generated L(X)
groups, which are X groups, the result is immediate because,

in this event, N(G)=S(G)cH, by hypothesis and Lemma 2.8.0

15

Definition 2.15: If X is a class of groups, then X satif-

 

fies the local theorem provided that we may conclude that

 

a group G is an X group whenever all the finitely generated

subgroups of G are X groups.

We observe that a class X satisfies the local theorem if and
only if all the subgroups of a group G are X groups whenever
all the finitely generated subgroups of G are X groups.

Another corollary to Lemma 2.11 is thus:

Corollary 2.16: SN satisfies the local theorem.

 

Many of the classes of groups that are commonly studied are
contained in SN. We have already cited the class of locally
Notherian groups. The class of all free groups is also con-
tained in SN since N(F)=S(F)=E if F is free on more than 1
generator, and N(F)=S(F)=F if F is free on 1 generator.
Hence, SN is the class of all groups in the event that homo-
morphic images of SN groups are again SN groups, since each
group is a homomorphic image of a free group. However, in
Chapter IV, we construct groups for which S(G) is a prOper

subgroup of N(G), so the class S is not quotient-closed.

N
It also follows that SN is not subgroup-closed since any
group can be embedded in a simple group (see [11], p. 316)
and each simple group is an SN group by Corollary 2.12. SN
groups are, however, closed under the taking of normal

subgroups.

16

Lemma 2.17: If G65” and K4 G, then KeSN.

Proof: N(K)=KnN(G)=KnS(G)cS(K) so S(K)=N(K) by Lemma 2.9.0

 

Duguid and Mc Lain [4] have proved that all FC-nilpotent
groups are locally Notherian, (hence belong to SN°) Thus,
for an FC-nilpotent group G, F1(G)cS(G). This is true

generally.

Proposition 2.18: For any group G, F1(G)CS(G).

 

Proof: Suppose xaFl(G), H is a non-finitely generated sub-

group of G, but <H,x> is finitely generated. Then

 

<H,x> H<x>H H .
H = —————fi—- :.______ is finitely generated so
<x> <x> <X>HnH

that <x>HnH is not finitely generated since H is not
finitely generated. But <x>H is a finitely generated FC—
nilpotent group, hence <x>HeMax so that <x>HnHeMax, a contra-

diction. Hence <H,x> is not finitely generated and xeS(G).0

CHAPTER III

THEOREMS

In this chapter we prove our main results, Theorems 3.10,

3.13 and 3.19.

Theorem 3.10 says that all polycyclic extensions of Abelian
groups are SN groups. Since s(AoP)=AoP, we need only show
that if GerP, then N(G)cH for all nearly finitely generated
subgroups of G (by Lemma 2.11.) Our first result in that
direction is Theorem 3.8, which gives sufficient conditions
for all normal Abelian subgroups to be contained in every
nearly finitely generated subgroup of G. That this is not
the case for an arbitrary group is shown in Theorem 4.1, in
the next chapter. The example constructed in Theorem 4.2
shows that even if all normal Abelian subgroups are con-
tained in all nearly finitely generated subgroups of G it is
not necessary that the locally Notherian radical be contain-

ed in each nearly finitely generated subgroup of G.

Theorem 3.13 gives sufficient conditions on a class X of
groups that the class of XoNilp groups is also contained
in S

N' The theorem applies in the case X=F, the class of

free groups.

17

18
Theorem 3.19 says that finite extensions of SN groups are

SN groups.

Our first lemma has as its corollary a key theorem, which
says that all normal nearly finitely generated subgroups

contain the locally Notherian radical.

Lemma 3.1: If X€N(G) and H<x> is a finitely generated

extension of H by <x>, then H is finitely generated.

Proof: Suppose the lemma is false and H is not finitely
generated. Since H<x>=<H,x> is finitely generated, Hn<x>=E,

and there exist hl,...,hneH such that <H,x>=<h1,...,hn,x>.

<>
x>

Hence H=<h1<X>,h2<X>,...,hn . Let

-1
h l , U =UluU1xuulx , and inductively

2

. = x x
define Un+1 UnuUn uUn , n>1.

Then <U,><<U, > for each i since H=u<U > and H is not
1 1+]. i J.

finitely generated. Hence, there exists heUl such that the

subgroup generated by the conjugates of h by all powers of x
lies outside of <Ui> for each i. We complete the proof by

constructing a non-finitely generated subgroup

G

Ac<[h,x],x>c<x> , which contradicts the fact that xeN(G).

19

(x)

Because h ¢<Ui> for all i, there exists a sequence

nl<n2<... of distinct non-negative integers such that

n

I ‘ n
i
hx e<Ui> but hX

l¢<Uj> for j<i. Let

xnk'l -1 xnk

- 2
1h" ,..., (h ) h >, k=l,2,... .

Ak=<h'1hx, (hx)

-1 xnk+1

Then Ak<Ak+l for all k since h h eA but

k+l'

—1 xnk+l
h h ¢<Uk>DAk. Let A=uAk. Then A is not finitely

k
generated since it is the union of a prOperly ascending

sequence of groups. Finally,

1 _ i+1 - i i
(hx ) lhx =(h 1h")x =[h,x1x e<[h,x],x>,
so that Akc<[h,x],x> for all k, and thus Ac<[h,x],x>,

which completes the proof.0

Theorem 3.2: If H is nearly finitely generated in G and

H4 G, then N (G)CH.

Proof: Deny the theorem and suppose xeN(G) but xtH. Then
<H,x> is finitely generated and <H,x>=H<x> since H<3G. By
Lemma 3.1, H is finitely generated, which contradicts that

H is nearly finitely generated in G.0

A particular case of the following technical lemma will be

of repeated use to us in proving Theorem 3.10.

20
Lemma 3.3: If Y is a subset of the group G with H=<Y>G

and <Y>4H, and if G/H=<ng|peR>, then G=<Y,gplpeR>.

Proof: If geG, then g=hw(g ,...,gp ) for some heH, w a
0l n

word and go ,...,go 6 [gplp€R}. Thus, for each XeY

l n

g hw(gp

x =x l'ooe'gpn) hW(gpl'ooe’g )

=(X ) 0n €<Y,gplpeR>
since <Y>4.H. Therefore H=<Y>GC<Y,gplpeR> and

G=<H,gplpeR>C<Y,gp|peR>.0

Corollary 3.4 is the form of Lemma 3.3 that we need., Its

proof is omitted since it is immediate from the lemma.

Corollary 3.4: If HOG, 5/1? is finitely generated, H is

 

is Abelian and not finitely generated, and if Y is a finite

subset, YcH, such that <Y>G=H, thenG is finitely generated.

Phillip Hall has proved the following theorem concerning

Abelian by polycyclic groups.

Theorem 3.5: ([5], p. 430, theorem 3) Every finitely
generated extension G of an Abelian group A by a polycyclic

group [=G/A satisfies Max-n.

In light of this theorem, our attention turns to the class
of groups satisfying Max—n, the maximal condition on normal

subgroups. Our next lemma is well known and says that in a

21
group satisfying Max-n each normal subgroup is finitely
normally generated.
G

k > for

Lemma 3.6: If GeMax-n and K4 G, then K=<kl,..., m

some k1,...,kmeK, m a positive integer.

Proof: If the lemma is not true, then we may construct a
properly ascending sequence

G

G
<k> <<k
1

,...,km> < ... < K

G
l,k2> < ... < <kl,k2
of normal subgroups of G contained in K. Since GeMax-n this
cannot happen: the chain must break off after finitely many

steps, and the lemma follows.0

Our next lemma gives us information about certain factori-

zations of finitely generated Max-n groups.

Lemma 3.7: If G=AH is a finitely generated Max-n group with

AQG, and A Abelian, then H is finitely generated.

Proof: Deny the assertion and suppose that H is not
finitely generated. Then AnH is not finitely generated
H AH G
since '——— : -— = - is finitely generated and H is not
AnH A A
finitely generated. Furthermore, AnH<IAH=G since A is
Abelian. We apply Corollary 3.4 with GEH and HéAnH and

conclude that

<a ,...,an>H<AnH for all a

1 'ooe'an6AnH' n21.

1

22

H AH . .
But <al,...,an> =<al,...,an> =<al,...,an>G, again Since A

G

is Abelian. Thus we have that <al,...,an> <AnH for all

al,...,aneAnH, n21. This contradicts the fact that G

satisfies Max—n, by Lemma 3.6. Hence, H must be finitely

generated.0

Theorem 3.8: If all the finitely generated subgroups of a
group G satisfy Max-n, then each normal Abelian subgroup A
of G is contained in each nearly finitely generated subgroup

H of G.

Proof: Suppose A<JG, A Abelian, and H is nearly finitely
generated in G but A¢H. Then K=AH is finitely generated and
thus KeMax-n by hypothesis. By Lemma 3.7, H is finitely

generated, a contradiction. Hence AcH.0

Our last lemma preliminary to proving Theorem 3.10 gives us
further information about factorizations in a more restrict-

ed setting than Lemma 3.7.

Lemma 3.9: If G=N(G)H is a finitely generated Max-n group,
and if A<IG, A Abelian, with G/AeMax and AcH, then H is

finitely generated.

Proof: If A is finitely generated, then GeMaxoMax so that

H is finitely generated. Suppose then that A is not

23

finitely generated. Since GeMax-n and A4 G,

A=<a ..,an>G for some a ,...,aneA, n21, (by Lemma 3.6)

l" l

=<al,...,an>N(G)H = N(G))H.

(<al,...,an>
Clearly ACN(G), and N(G)/AeMax since G/AeMax so that

NAG) = <Axl,...,Axk> for some xl,...,xkeN(G), k21.

N(G)
Hence <a ... a > c <a ... a x ... x > = K
I ll I n II I nI lI I 1"

since <al,...,an>cAcN(G) and A is Abelian. Now KcN(G) so

. . . . G
KeMax Since K is finitely generated. Thus <al,...,an>N( )

N(G)

is finitely generated and <a1,...,an> =<b ,...,br> for

1

H

some b breA,r21. Hence A=<bl""’br> . Now A4H and

1,000,
H/A is finitely generated since G/AeMax. Corollary 3.4

applied to GSH and HEA yields that H is finitely generated.0
Theorem 3.10: AoPcSN

Pgoof: Since S(AoP)=AoP, it suffices by Corollary 2.13 to
Show that GerP implies N(G)cI(G) . Thus, suppose that GerP,
H is nearly finitely generated in G but N(G)¢H. Then
K=N(G)H is finitely generated and K=N(K)H since N(K):N(G).
Also, H is nearly finitely generated in K. Now K is a
finitely generated AoP group so that KeMax-n by Theorem 3.5.
Let A<IK, A Abelian, with K/AeMax. Then AcH by Theorem 3.8,
and Lemma 3.9 yields that H is finitely generated. This

contradicts that H is nearly finitely generated in G, so it

24

must be that N(G)cH, and the theorem is proved.0

Since SN satisfies the local theorem (Corollary 2.16), it is
immediate that L(AoP)cSN. From this observation follows the
inclusion in SN of the subclasses of L(AoP) described in

Chapter I.

Theorem 3.11: The classes AoNilp, AoL(Nilp) and L(AoNilp)

are contained in SN'

Proof: This follows immediately since AoNilpcAoL(Ni1p)c

L(AoNilPICL(AoP)CSN.0
We now investigate the circumstances under which a class X
has the prOperties:

(i) S(XoNilp)=XoNilp, and (ii) XoNilpcSN.

In pursuing this question it is convenient to make the

following definition. Suppose X is a class of groups.

Definition 3.12: X satisfies (*) if GeXoNilp, Kq G, KeX

 

with G/KeNilp implies that N(K)=KnN(G)cH for all nearly

finitely generated subgroups H of G.

Returning to our question it is clear that we satisfy (i) if
(iii) S(X)=X. Suppose that X satisfies (ii) in addition to

(i). If GeXoNilp, K<G with KeX and G/KeNilp, then

25
certainly KnN(G)cH whenever H is nearly finitely generated
in G since N(G)=S(G)cH by Lemma 2.8. Thus X satisfies (*).
In fact the converse of this statement is true, and its

proof gives us our result.

Theorem 3.13: If S(X)=X, then XoNilpcSN if and only if X

has prOperty (*).

Egoof: It only remains to Show that if X satisfies (iii) and
(*), then XoNilpcSN. Under these assumptions we have al-
ready commented that s(XoNilp)=XoNi1p so that by the remark
following Corollary 2.13 we need only Show that if GeXoNilp
and H is nearly finitely generated in G, then N(G)CH. So

let GeXoNilp, KQG, with KeX and G/KeNilp. Then G has the
invariant series

E4K=Zod 214 ...d Zn=G, for some n20,

where the 21 are from

K/K=Zo/K 4 Zl/K 4 . . . 4 Zn/K=G/K,

the upper central series for G/K. We show that N(G)nzicH
for i=0,l,...,n. The proof is by induction, and the case

i=0 is simply the assumption that X satisfies (*).

Suppose now that N(G)nzicH for some i, Osisn-l. We claim

that N(G)nzi+1 normalizes H. Let er, y€N(G)nZ Then

i+l°
[x,y]€ZinN(G)CH by the inductive hypothesis. So,

1

[x,y]=x— xYEH, which gives xYeH since er. Thus we have

26

that H¢:[Zi+lnN(G)]H=L, and H is nearly finitely generated

in L since H is nearly finitely generated in G. By Theorem

3.2, H3N(L)Dzi+lnN(G), which completes the induction.

Hence, HDZnnN(G)=N(G), and the proof is complete.[]

The question of the existence of non-trivial subgroup-closed
classes X satisfying property (*) is answered in the next

lemma.

Lemma 3.14: If S(X)=X and GeX implies N(G)=E or GeA, then

X satisfies property (*).

_P_r_o_c£: Suppose GeXoNilp, K4 G, KeX with G/KeNilp and
suppose H is nearly finitely generated in G. Now
KnN(G)cN(K). If KnN(G)=E, then KnN(G)cH is trivial. If
KnN(G)>E, then N(K)>E so that KeA, by hypothesis. Hence

GerNilpcsN so that KnN(G)cs(G)cH.0

Example 3.15: The class X=A, of Abelian groups, satisfies
(*) by Theorems 3.11 and 3.13. A more interesting case is
obtained by choosing X=F, the class of all free groups. To
see that Lemma 3.14 applies, we recall that subgroups of
free groups are free, and 1—generator free groups are cer-
tainly Abelian. Thus it remains only to show that N(F)=E if
F is free on more than 1 generator. Suppose K<3F, K>E; K
cannot be Abelian so that K is free on more than 1 generator.

Let H be a 2-generator subgroup of K. Then H is free and

El

27
IH:H'|=°° so that H' is not finitely generated (see [8],
p. 104.) Hence N(F)=E and the hypotheses of Lemma 3.14 are
satisfied. Therefore, F satisfies prOperty (*) and Theorem

3.13 yields FoNilpCSN.

Our attention now turns to proving that finite extensions

N groups. We begin with a lemma that

gives an easy rule for identifying some of the non-finitely

of SN groups are 5
generated subgroups of a group.

Lemma 3.16: If HcG, H is not finitely generated, and

HCHcHS(G), then H'is not finitely generated.

Proof: Suppose to the contrary that H is finitely generated.

Then there exist $1,...,sneS(G) such that HS<H,Sl,...,sn>.

Let H =H and inductively define Hi+

0 =<Hi,s >, 05i<n.

1 1+1
Then H0=H is not finitely generated by hypothesis, and if
we assume that H1 is not finitely generated, then

H. =<H.,S
1

1+1 > is not finitely generated Since si+leS(G)

1+1
for each i. Thus HéHn is not finitely generated, a

contradiction.0

In any group G if K<JG, then N(K)=KnN(G), so that in partic-
ular N(K)cN(G). If finite extensions of SN groups are
again to be SN groups, then we must certainly have that

S(K)cS(G) for normal subgroups K of finite index in G.

28

Lemma 3.17: If KQG and lG:K|<co , then S(K)cS(G).

Egoof: Deny the lemma and suppose that xeS(K)-S(G), H is a
non-finitely generated subgroup of G, but <H,x> is finitely
generated. Now lH:HnKl=|HK:KISIG:KI<¢n so that HnK is not
finitely generated since H is not finitely generated.
Similarly l<H,x>:<H,x>nK|=l<H,x>K:K|SIG:Kl<¢»so that <H,x>nK
is finitely generated since <H,x> is finitely generated.
Furthermore, <HnK,x> is not finitely generated since xeS(K)
and HanK is not finitely generated, and clearly

HnK c <H,x>nK. To complete the proof, we show that
<H,x>an(HnK)S(K) giving Han<H,x>nKC(HnK)S(K). By Lemma
3.16 we find that <H,x>nK is not finitely generated, which

is a contradiction. Suppose then that ge<H,x>nK=(H<x>H)nK.

Then g=h§ for some heH, §t<x>H. But <x>HcS(K) Since
an(K)0K4 G, so FieK. Now, we have that g,§€eK so that

heHnK. Hence g=h§e(HnK)S(K), which completes the proof.E]

Our last lemma is of the exercise variety and, although we
cannot remember having seen it, we suSpect that it is prob-

ably well known.

Lemma 3.18: If A, B, C, D, BD and AC are subgroups of G

with CQBD, and if IB:A|, ID:C|< no, then IBD:ACI< 09.

Proof: Suppose that Ax1,...,Axn form a complete set of

right cosets of A in B, and Cyl,...Cym form a complete set

 

29
of right cosets of C in D. Let g=bdeBD, beB, deD. Then

bd=(axi)(cyj) for some aeA, ceC, lSiSn, lsjsm. Since C<IBD
xic=cxi for some ceC so that bd=(axi)(cyj)=a(xic)yj=acxiyje
(AC)xiyj. Hence, (xiyjllsisn, lSanl] includes a complete

set of right coset representatives of AC in BD, whence

|BD:AC|Snm<°° . D

Theorem 3.19: Finite extensions of SN groups are SN groups.
Proof: Suppose Kq G, KeSN, lG:K|<oo , and let xeN(G). We
prove that xeS(G). If xeK, then xeKnN(G)=N(K)=S(K)cS(G)

by Lemma 3.17. Suppose then that x¢K and x¢S(G). Then

there exists a non-finitely generated subgroup H of G such

that <H,x>=H<x>H

is finitely generated. As in the proof of
Lemma 3.17 IH:HnKI<¢: and HnK is not finitely generated, and

a similar argument gives |<x>H:<x>HnK|<w». Furthermore,

<x>HnK4H<x>H so that (KnH) (<x>HnK)cH<x>H. Applying Lemma

3.18 with A=KnH, B=H, C=<x>HnK and D=<x>H yields that

|H<x>H: (KnH) (<x>HnK)I<oo so that (KnH) (<x>HnK) is finitely

generated since H<x>H is finitely generated. But

<x>HnKC<x>GnKCN(K)=S(K) so that by Lemma 3.16 (KnH)(<x>HnK)

is not finitely generated since KnH is not finitely genera-
ted. This gives our contradiction so that xeS(G), and the

proof is complete.[]

30
An easy extension of Theorem 3.19 is that if all the sub-
groups of a normal subgroup L of finite index in G are SN
groups, then all the subgroups of G are SN groups.

Corollary 3.20: If K<1G, IG:K|<~ and LeS for all LcK,

N

 

then fies” for all LtG.

Proof: Let L—CG. Then LanL and IL:L-nK|=IK'l—.:KIS|G:K|<~ .

By hypothesis, then, LnKeS Since IL:LnKI<~ , Theorem 3.19

N.
gives LeSN.E]

CHAPTER IV

EXAMPLES

In this chapter we construct groups G whose nearly finitely
generated subgroups H or whose subgroup S(G) possess remark-
able prOperties. The primary examples are of groups that
lie outside the class SN'

A general strategy for constructing such groups is to find a
finitely generated group G with a non-finitely generated
subgroup H such that <H,N(G)>=G. If G is such a group, then
there exists a nearly finitely generated subgroup H’of G
containing H, and H clearly cannot contain N(G), so that we
must have S(G)¢N(G) (since in any group we always have that
S(G) is contained in every nearly finitely generated sub-

group, by Lemma 2.8.)

All our constructions are semi-direct products (Split
extensions) and all but one of these are wreath products.

We recall here the structure of semi-direct products and
wreath products. For details and proofs concerning semi-
direct products see Scott ([11], section 9.2), and for wreath

products see Neumann [9].

31

 

 

 

 

32

A group G is the semi—direct pooduct of its subgroups H and

 

K if and only if H<IG, G=KH, and KnH=E. In this case, every
element of G has a unique representation as kh, with keK and

heH. Multiplication in G is given by (klhl)(k2h2)=

k
(klk2)(hl 2hz). If T denotes the inner automorphism of G

k

induced by keK (that is, ng=gk for all geG), the function T
defined by kT=TkIH is a homomorphism from K into Aut(H).

Multiplication in G now becomes

(klhl)(k2h2)=(klk2)((hl(k2T))h2). In this event, G is a

semi-direct product with (associated) homomorphism 1.

Conversely, suppose we are given two groups H and K and a
homomorphism T:K+Aut(H). Let G=KxH as a set, and define
multiplication in G by

(k1,h1)(k2,h2)=(k1k2,(h1(k2T))h2). Under this multi-

plication, G is a group. Let K*= ((k,e)|keK l,

H*= [(e,h)lheH I. Then K* and H*are subgroups of G. Define
U and V by hU=(e,h), kV=(k,e), heH and keK. U and V are
isomorphisms of H onto H* and K onto K*, reSpectively. Let
T* be the homomorphism from K* into Aut(H*) that satisfies
(hU)((kV)T*)=(h(kT))U. Then G is the semi-direct product of

*
H* by K with homomorphism T*.

Because of the natural correspondence between H and H*,
* * .
K and K , and T and T , we will say that G is the semi-

direct product of H by K with homomorphism T.

33
In our examples, we will be interested in the case where K
is a subgroup of Aut(H) and T is the embedding of K into
Aut(H). Multiplication then becomes
(al,hl)(02,h2)=(a1a2,(h1a2)h2), h1,h2€H, al,a26K. In
this instance, G is a relative holomorph of H by K, written

G=Hol(H,K).

 

We identify (a,e) with a and (e,h) with h, aeK, heH. If we
compute [h,a], then on the one hand we get h-lh“ and on the
other we get

1 -l

[h,a]=h- a hot=(e,h-l

)(a'1.e) (e,h) (one)
=(a-1,h-la-l)(e,h)(a,e)=(a-l,(h-la-l)h)(a,e)
=(e,<(h‘la'1)h)a)=(e,h‘1(ha))
=h-1(ha).

Hence, we have that the action of a on h, ha, is given by

he

We are now ready to define the standard restricted wreath
product, AIB, of the group A by the group B. Let
K={f|f:B+A and f(b)=e for only finitely many beBJ
Then K is a group under ordinary componentwise multiplica-
tion, and each element beB induces an automorphism 5 of K
defined by
f5(b')=f(b'b-1) for feK, b'eB.

If Bé{31beB], then B becomes a group under function

34
composition 5i3é= 152, and B'is clearly isomorphic to B

under the correspondence b+3. Thus, we identify B with B
and consider B to be a subgroup of Aut(K). -Finally, we
define A1B by

A1B=H01(K,B).

 

It is customary to call K the base group of the wreath

 

product and B the top group. Thus, A1B=KB, with K<1AIB,and

 

KnB=E.

For each beB, there correSponds a subgroup Kb of K isomor-

phic to A. Kb is defined by
Kb=[lfeKlf(x)=e if xxb‘].

Thus, K is isomorphic to the direct sum of lBl copies of A.
We identify A with the subgroup Ke of K, where e is the
identity in B. Hence AIB=<Kb,B> for each beB and A13 is
finitely generated if and only if A and B are finitely

generated.

Theorem 4.1: SN is a prOper subclass of the class of all

groups.

Egoof: Let B be any finitely generated group, BéMax, and
let C be nearly finitely generated in B. Then IB:C|=¢”
21B. We Show
Ithat G23”. Let K be the base group of the wreath product

because C is not finitely generated. Let G=Z

and let

35
M: [feKlf has an even number of non-zero entries on
each right coset of C in B').
M is clearly a subgroup of K, and B normalizes M since the
members of B Simply permute cosets of C in B. Furthermore,

M is not finitely generated, for, if [x ,...,x ,... ) is a

l n

complete set of coset representatives for the right cosets

of C in B, with xl=e, and if

Mi=[fele(x)=0 for all xerj, j>i],

then Ml<M2<...<Mn<... and M=uMi, so that M is a union of a
i

prOperly ascending chain of subgroups and thus cannot be

finitely generated.

Moreover, MB is not finitely generated, for, suppose to the

contrary that MB=<fl,...,fn,bl,...,bm> with f fneM and

1'...’
b1,...,bmeB. Now without loss of generality, we may assume

that fieMl for each i since Mc<Ml,B>. For each ceC,c¢e, let
1 if x=c or x=e
be given by fc(x)= . Then with-

let fceM
0 otherwise

1

out loss of generality, fi=fc- for some cieC, i=1,...,n,
i

1 if X6 (Xi ,..OIXi )
for, if fi(x)= 1 2n (with the conven-

0 otherwise

tion that xi =0 if 06 {xi ,...,xi } ), then

1 1 2n
f -f - -f if x- 20'
X- X- X- l
f - 11 12 lzn l
1 f ...-f ifx.=0 '
x. x 11

36

Now if Ce<cl,...,cn> and CeC-C, then we claim that

b

C .
1.

fC¢<f1,...,fn,bl,...,bm>. For, if bEB, then f has 2 non-

zero entries on the coset of E in B containing b so that

the product of conjugates of the fc has an even number of
i

non-zero entries on each coset of E'in B. Hence,

fC¢<fl,...,fn,bl,...,bm> so that MB is not finitely genera—

ted.

Now, <K,MB>=<K,B>=G and KcN(G) so that G=<N(G),MB> with MB
not finitely generated, so that S(G)=N(G) by the introductory
remarks to this chapter. Hence, GtSN.[]

Theorem 3.11 says that AoNilp is contained in S Our next

N‘
example shows that the containment does not hold for NilpoA.
In fact, we construct a group G which is an Abelian extension
of a free nilpotent group of class 2, but is not an SN group.
G, of course, is an extension of an SN group by an SN group,

and thus SNoSN¢SN. (The same conclusion could be drawn from

the construction in Theorem 4.1.)

If we define the S-series of a group G by (i) SO=E, and
(11) Si+l is that subgroup of G such that Si+l/Si=S(G/Si)'
i20, and if we define the N-series by (i) N0=E, and

(ii) N1+1 is that subgroup of G such that Ni+l/Ni=N(G/Ni)’

37
i20, then the G in our next example has the prOperty that
82<S3=G while N1<N2=G. Thus, the S and N series may both

reach G though not necessarily in the same number of steps.

Before proceeding with the construction, we recall some
definitions and a few of the results that we need pertaining
to free, free nilpotent, and free Abelian groups. Details

may be found in [8].

If F is free on the set S=[ ...,x_l,x0,x1,...} and w is a

word (that is, weF), then w is a law in the group G if the
only possible value of w in G is e, the identity. By this
we mean that for every homomorphism a:F+G wa=e. (For

example, [xi,xj] is a law in every Abelian group.)

A group K is free Abelian if K is isomorphic to F/F' for

some free group F. A group G is free nilpotent if G is iso-

 

morphic to F/[n(F) for some free group F and integer n21.
Every Abelian group is a homomorphic image of a free Abelian
group and every nilpotent group of class n is a homomorphic
image of a free nilpotent group of class n. If G is a free
nilpotent group of class 2, then G'=Z(G), and if G is any
free nilpotent group and TCG is a subset containing at least
2 elements, the T freely generates a free nilpotent group if
and only if T is linearly independent modulo G' (see [8],

p. 121.)

38

Theorem 4.2: NilpoA¢S S 03 ¢S

N'NNN'

Pgoof: Let X=(...,x_1,xo,x1,...} be a countably infinite

set indexed by the integers and F=F(x) be the free group on
X. Consider the group G=<K,a> where K=F/r3(F) is the free

nilpotent group of class 2 on X and a is the Neumann shift

operator defined by a:xi[3(F)+xi+1f3(F). (To simplify the

notation we will omit the [3(F) in writing members of K.)

Now aeAut(K) and G=Hol(K,<a>), so that G is a split extension
of K by <a>. Hence, G=K<a>eNilp°A, whence KcN(G). Note

also that <K,a>=<xo,a>, which says that G is finitely gener-
ated. To Show that GKSN we need only find a non-finitely
generated subgroup H of G containing a, for then G=<K,a>=
<N(G),H> and S(G)¢N(G) by our comments at the beginning of

this chapter. To this end consider the subgroup H=<K',a>.

Now K'=Z(K) and K' is free Abelian, freely generated by the

set Ké{[xi,xj]|i<ji, Since (i) K surely generates K', and

(ii) ifZ n [xi ,x. ]=0, then every nilpotent group of class
k k k 3k

2 on, say, the 2 generators xil and le satisfies the law
nl

[xil,xj1] , which is certainly not the case. Suppose now

that <K',a> is finitely generated. Without loss of general-

ity, then, <K',a>=<[xo,xill,...,[xo,xin],a>, x. >0,

13'

j=1,...,n, so that

K'=K'n<K',a>=<[xi,xj]lj-ie{i1,...,in[> <K', a

39

contradiction.

Thus far we have shown that S(G)¢N(G). To prove that N(G)=K
it remains only to show the inclusion N(G)cK. To obtain
this, we observe that if M>K, then ameM for some integer mzo

(without loss of generality m>0). Hence <K,am>=

(XOIXl,---.Xm_l,am> is finitely generated and

lG:<K,am>|=m so that IG:MlSm and thus M is finitely generated

since G is. Since K<M M¢L(Max) so that N(G)=K.

To Show that Z(K)CS(G), suppose to the contrary that 25Z(K),

HcG is not finitely generated, but <H,z> is finitely
H

 

 

generated. Then H¢K (since KeL(Max)). <H,z>=H<z> and
<H,z> H<z>H ~ H . . .
= - ——————- is finitely generated so that
H H H
<z> <z> <z> nH

<z>HnH is not finitely generated since H is not finitely

generated. Thus, H=< <z>HnH,hl,...,hn> for some h1,...,hneH.

If heH, then h=amk for some integer m, keK, and

m m

m
h amk a )k=z“ since 2“ eZ(K)-

2 =2 =(z
Hence, there exists an infinite subset Icz such that
H oi - mi
<z> nH=<z I1€I>CZ(K) and hi=a ki for some mieZ, kieK,
lSiSn. Furthermore, mi¢0, and without loss of generality
m1>0. But then,

i m
< <z>HnH,hl>=<z“ ,a lkllieI> is finitely generated;

40

for, if we choose a subset Im cI that contains a unique
1

representative of every coset of <ml> in Z that is found in

I (that is, for each ieI, i=j mod m1 for some jeIm , and
l

jxl’ mod m1 if i=6". jJeIm ). then
1

- m
<<z>HnH,hl>=<zaj,a lklljeIm> is finitely generated
1

since Im contains at most m1 members. Hence,
1

3
h >=<za ,h ,...,h IjeI > is finitely
n l n m1

H=<<z>HnH,hl,...,

generated, a contradiction. Hence, 265(6).

Suppose finally that xeK-Z(K). We Show that x¢S(G). The

construction follows the lines of the early part of this

m
proof. Let L=<x° Imez>. Then LcK and L is a free nilpotent

am

group of class 2 freely generated by the x , mez, since

m
[x“ ImeZ) is clearly linearly independent modulo K'. For

convenience, let yi=xai for each i. Then L=<...,y0,y1,...>,
L4 <L,a>, and a simply permutes the y...L in the same manner as
the xi. AS earlier in the proof, we may conclude that
<L',a> is not finitely generated, but

<<L',a>,x>=<L',a,y0>=<yo,a> is finitely generated so

that X¢S(G).

41
The lengths of the S and N series may easily be found.

N(G)=K and G/K:<a> so that N(G/K)=G/K and N =G.

2
S(G)=Z(K)=K' so that G/S(G)=G/K'erA. But then,
S(G/S(G))=N(G/S(G))=N(G/K')=K/K' (by Theorem 3.11.) There-

fore, SZ=K and S(G/S2)=S(G/K)=G/K since G/KeA. Hence $3=G.

In the introductory comments to Chapter III we remarked that E
G in this example has the prOperty that all the normal
Abelian subgroups of G are contained in I(G), yet N(G)¢I(G).
We have already seen that N(G)¢I(G), since <K',a> is con-
tained in some nearly finitely generated subgroup H of G,
and N(G)¢H since <N(G),H>:<N(G),a>=G. To see that all the
normal Abelian subgroups of G are contained in I(G) we Show

that if A<)G, A Abelian, then AcZ(K)=S(G)cI(G), by Lemma 2.8.

S
Suppose x=(ar,k)eA, r an integer, keK, and let y=x° , S an

integer. A straight-forward computation Shows that

-l s+r
a

[x,y]=k-1(k )(kar)(kas). The hypothesis that A is a

normal Abelian subgroup of G requires that [x,y]=e, which

is equivalent to (kar)(kas)=(kas+r)k. Since a is the
Neumann Shift Operator, the only way this equality can hold

if s¢0 (xry) is for r=0. Hence, we must have =k and

k(kas)=(kas)k. This latter equality holds only if
keZ(K)=K'=S(G). Since k is arbitrary, AcZ(K)=S(G), as

required.[]

42
Our next example shows how to construct a class of groups
having nearly finitely generated subgroups that are not
normal. However, we first need some preliminary information
about certain subgroups of Z IZ.

2

Definition 4.3: If fEKCAIZ=KZ, then f‘i_ oo [r,s], r and s

 

integers, if f(zm)=e for all m not between r and s inclusive.

Lemma 4.4: If G=Z 12=KZ and H>Z, then ngzlz and |G:H|<O ,

2

and if L>K, then L is finitely generated.

Pgoof: Since H>Z, HnK>E, and since Z may be used to Shift
the interval that a function is on, as per the previous
definition, we choose feK such that f is on [0,n], n>0, with
n as small as possible. We claim that H=<f,z>, for suppose
that g is a function in H, g is on [k,r]. Then, by multiply-
ing g by apprOpriate conjugates of f (by powers of 2), we

may reduce the length of the interval that g is on until we
finally arrive at a function E that is on [0,n]. But 3 must
be f since if it were not, then 3f would be a non-trivial
function on [0,5] for some 0<Hkn, which contradicts our

l§e<f,z> so that

choice of f. Hence, we have that g and g'
ge<f,z> and H=<f,z>. Incidentally, we have also shown that
f is unique in H with reSpect to being on [0,n].

1 if n=0

Let gEK be the function defined by g(zn)= .
0 if n10

43

n n n n
Then G=<g,z>. The mapping (zk,gz l...gz r)->(zk,fz 1...f2 r)

is easily checked to be an isomorphism of G=221Z onto

H=<f,z>. Hence, szziz.

To Show that |G:H|<", let yeG. Then Hy=ngr for some gEK,

r an integer, so Hy=Hz~rgzr=H§, EEK, since zeH. AS we
argued before, we may multiply g’by apprOpriate conjugates
of f, all of which are in H, until we get a function 3‘, and
3' is on [0,n]. Thus, Hy=H§‘, with 3‘ on [0,n]. Since the
number of functions in K on [0,n] is finite we have shown
that H has only a finite number of right cosets in G so that

lG:H|<co .

Finally, if L>K, then L=K<zs> for some positive integer s

and L:(Zz+...+zz)12 by [9, Lemma 8.1]. Thus L is finitely
s-c0pieS ‘

generated.[]

Theorem 4.5: If A is finitely generated and G=A1(ZZIZ)=
K(Z212) , then K2 is a non-normal nearly finitely generated

subgroup of G.

Proof: G is clearly finitely generated since A and 2212

are finitely generated. Suppose H>H=Kz. Then

G H H H
212 g - a - > - : Z. Hence, - is isomorphic to a subgroup
K K K K

Z

44
of ZZIZ that prOperly contains the tOp group Z. Thus, by
Lemma 4.4, lG/K:H/K|<O so that IG:H|<¢' and H'is finitely

generated since G is finitely generated.

That H is not normal in G follows from the fact that the
normal closure of Z in 2212 is larger than Z, so that the
normal closure of H in G is greater than H.

If N is the base group in Z 12 and Ké{feK|f(x)=e if x¢N],

2
then certainly KceK,Z> so that if H=KZ is finitely generated,

say H=<f fn,z>, f1,...,fn5K, then without loss of

1,000,

generality, fl,...,fneK. But then,
Ké<fl,...,fn,z>nKé<f1,...,fn>. This says that K is

finitely generated, which is impossible since N is not
finitely generated. Hence, H is not finitely generated, and

the proof is complete.0

Our next example shows that it is not the case that the
intersection KnH of a nearly finitely generated subgroup

H of G with a normal subgroup K of finite index in G need be
nearly finitely generated in K. First we need a lemma

concerning the subgroups of 212 that contain the tOp group.

Lemma 4.6: If H contains the top group in 212, then H is

finitely generated.

45
Pgoof: If K is the base group in 212, then H=(HnK)Z and HnK
is normal in leo le is a finitely generated Abelian
extension of an Abelian group, so 212 satisfies Max-n by
Theorem 3.5. Hence HnK is finitely normally generated so

that H=(HnK)Z is finitely generated.0
Example 4.7: There exists a group G containing subgroups
H and K, with K<G, |G:Kl<¢I and H nearly finitely generated

in G, but HnK is not nearly finitely generated in K.

Construction: Let G=AIZ where A is any finitely generated

 

group containing a nearly finitely generated subgroup B.
Let us identify B with the subgroup of the base group K
consisting of all functions f satisfying
e if n¢0
f(z“): . If L={feKlf(zn)eB for all n}, then
beB if n=0
<B,Z>=LZ:BIZ, Z normalizes L and L2 is certainly not finitely
generated since B is not finitely generated. Consider the

subgroup K<zz>. Clearly IG:K<zZ>|=2, so K<zz><lG, and

LZn(K<zz>)=L<zz>.

L<zz> is not nearly finitely generated in K<zz>, for consider

the subgroup M of G defined by

2n+1

M={feKlf(zzn)eA, f(z )eB, n an integer}. Then <zz>

2

normalizes M, L<z ><M<22> and M<22>is not finitely generated

since B is not finitely generated.

46
Thus, it remains only to Show the existence of an A such
that L2 is nearly finitely generated in G. To this end,
consider AéfztiéKE. K'is not finitely generated but K’is

nearly finitely generated in A by Lemma 4.4 Now, we have

- _ _. _ __ Z _Z_Z _
G=(ZZIZ)IZ=(ZZIZ)ZZ=(KZ) Z:(K Z )Z, L=Kz and LZZKZZ.

Also, KZ<IG since fa as well as Z, normalizes KG Now,

(x—z') zz (rz'z'z) z n

—z _ —-z : ZZZ= 212. Thus, if H>Lz:i<'zz, then :-
K K K

KZ

 

G
:5 =
is isomorphic to a subgroup of 712 containing the tOp group,
Z, and so is finitely generated by Lemma 4.6. So, we have
_—Z =‘—z 2.2
that H—<K Z,gl,...,gn> <K ,z,gl,...,gn> for some g1,...,gne ,
and without loss of generality, we may assume that 91(1)¢e,

for all i, since we may conjugate the 91 by powers of 2, if

necessary, to effect this condition. Furthermore,
..z ..z —z _z
K Z=<K1,z> so that H=<K1,z,g1,...,gn>. Now <K1,gl>

is finitely generated since gl(l)¢e so that <K§,gl> is

isomorphic to a subgroup of iéIZEKE'containing'the base
group K and thus is finitely generated by Lemma 4.4. Hence,

H is finitely generated, and the proof is complete.0

Our last example is a counterexample to a conjecture made by
J.E. Roseblade, in discussion. The conjecture was that for
polycyclic extensions of Abelian groups, G, S(G)=N(G)=

Fitt(G), where Fitt(G), the Fitting subgroup of G, is the

 

47
maximum normal nilpotent subgroup of G (which exists for any
Max-n group.) We have already seen that the first equality
in the assertion is true (Theorem 3.10), but the following
example shows that Fitt(G) can be a preper subgroup of N(G)

if GEAOP.

Example 4.8: In G=Zp1(quZ), where p and q are distinct

primes, Fitt(G)=K, the base group, while S(G)=N(G)=qu.

Proof: KZq is clearly normal in G and locally N5therian,
but not finitely generated. If H>KZq, then IG:H|<¢’ so

that H is finitely generated. Thus H cannot be locally

N5therian, and it follows that N(G)=qu.

We recall that in AIB=KB, if B :8, then KB is isomorphic

l 1

to the wreath product of a direct sum of a certain number

of c0pies of K with B ([9, Lemma 8.11), and that A1B is

1

nilpotent if and only if A and B are p-groups for the same
prime p, B is finite and A is of finite exponent (see [3].)

Now K is a p-group and if er xZ, then x is either of in-

q
finite order or of order q. Thus K<x> cannot be nilpotent

since pzq. Hence K, which is Abelian, is the Fitting sub-

group of G.[]

BIBLIOGRAPHY

10.

ll.

BIBLIOGRAPHY

Baer, R. "Lokal Noethersche Gruppen," Math. Z.
66 (1957), 341-63.

Baer, R. "Group theoretical prOperties and functions,"
Colloq. Math. XIV (1966), 285-327.

Baumslag, G. "Wreath products and p-groups,"
Proc. Cambridge Philos. Soc. 55 (1959), 224-31.

Duguid, A.M. and McLain, D.H. "PC-nilpotent and FC-
soluble groups," Proc. Cambridge Philos. Soc.

Hall, P. "Finiteness conditions for solvable groups,"
Proc. London Math. Soc. (3) 4 (1954), 419-36.

Kurosh, A.G. The Theor of Grou 5, Vols. I and II.
New York, Che sea, I560.

Magnus, W., Karass, A. and Solitar, D. Combinatorial
Group Theogy: Presentations of rou s in terms of
generators and relations, PuEE and AppIIed .
Mathematicst—VbI. XIII, New York, Interscience,
1966.

 

Neumann, Hanna. Varieties o; Grou s, Ergebnisse der
Mathematik and ihrer GrenzgeEiete (N.S.), Band
37. New York: Springer-Verlag, 1967.

 

Neumann, P.M. "On the structure of standard wreath
products of groups," Math. Z. 84 (1964), 343-73.

Riles, J.B. "The near Frattini subgroups of infinite
groups," J. Algebra 12 (1969), 155-71.

Scott, W.R. Grou Theor , Englewood Cliffs, New Jersey:
Prentice-Ha 1, I§6§

48

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